A factorization procedure for finite algebras

Abstract

A procedure is developed for decomposing any finite algebra into a minimal set of maximally independent simple homomorphic images, or factors, of the algebra. The definition of admissible sets of factors is made in relation to the congruence lattice of the algebra, and generalises the notion of an irredundant reduction in a modular lattice. An algorithm for determining all possible sets of factors of a given finite algebra is derived and an index for measuring the degree of independence of factors is defined. Applications of the technique to finite algebraic models within the social psychological domain are presented and include factorizations for certain semigroups of binary relations and for a class of finite semilattices.